
where \(W_t = \sqrt{\kappa } B_t\)  and \(B_t\)  is a standard Brownian motion. The solution to (REF ) exists until the time \(T_z = \inf \lbrace t >0: \mathrm {Im}(g_t(z)) = 0 \rbrace \)  and we write \(K_t = \lbrace z \in \mathbf {H}: T_z \le t \rbrace \) . Then \((g_t)_{t \ge 0}\)  is a family of conformal maps called the \(\mathrm {SLE}_\kappa \)  Loewner chain and \(g_t : \mathbf {H}\setminus K_t \rightarrow \mathbf {H}\)  is the unique conformal map with \(g_t(z) - z \rightarrow 0\)  as \(z \rightarrow \infty \) . It was proved by Rohde and Schramm for \(\kappa \ne 8\)  [1]} and by Lawler, Schramm, and Werner for \(\kappa = 8\)  [2]} that there a.s. exists a continuous curve \(\eta \)  in \(\overline{\mathbf {H}}\)  from 0 to \(\infty \)  such that \(\mathbf {H}\setminus K_t\)  is the unbounded component of \(\mathbf {H}\setminus \eta ([0,t])\) . The curve \(\eta \)  is an \(\mathrm {SLE}_\kappa \)  process from 0 to \(\infty \)  in \(\mathbf {H}\) . The behavior of \(\mathrm {SLE}_\kappa \)  processes depend heavily on \(\kappa \) : when \(\kappa \le 4\) , they are a.s. simple, when \(\kappa \in (4,8)\)  they have self-intersections and intersect the domain boundary and when \(\kappa \ge 8\)  they are a.s. space-filling [1]}.
